On the relations between a singular system of differential equations and a system with delays

Year 2022, Volume: 2 Issue: 4, 221 - 227, 30.12.2022

Ioannis Dassios

https://doi.org/10.53391/mmnsa.2022.018

Cited By: 2

Abstract

In this article, we consider a class of systems of differential equations with multiple delays. We define a transform that reformulates the system with delays into a singular linear system of differential equations whose coefficients are non-square constant matrices, and the number of their columns is greater than the number of their rows. By studying only the singular system, we provide a form of solutions for both systems.

Keywords

Delays, differential, singular systems, underdetermined systems

References

• Batiha, I., El-Khazali, R., AlSaedi, A., & Momani, S. The general solution of singular fractional-order linear time-invariant continuous systems with regular pencils. Entropy, 20(6), 400, (2018).
• Campbell, S.L. Singular systems of differential equations, Pitman, San Francisco, Vol. 1, 1980; Vol. 2, (1982).
• Dai, L. Singular Control Systems, Lecture Notes in Control and information Sciences Edited by M. Thoma and A. Wyner (1988).
• Dassios, I.K., Zimbidis, A., & Kontzalis, C.P. The delay effect in a stochastic multiplier–accelerator model. Journal of Economic Structures, 3(7), 1-24, (2014).
• Dassios, I., Tzounas, G., & Milano, F. The Möbius transform effect in singular systems of differential equations. Applied Mathematics and Computation, 361, 338-353, (2019).
• Dassios, I. Stability and robustness of singular systems of fractional nabla difference equations. Circuits, Systems and Signal Processing, 36(1), 49-64, (2017).
• Dassios, I., Tzounas, G., & Milano, F. Participation factors for singular systems of differential equations. Circuits, Systems and Signal Processing, 39(1), 83-110, (2020).
• Dassios, I., & Kalogeropoulos, G. On a non-homogeneous singular linear discrete time system with a singular matrix pencil. Circuits, Systems, and Signal Processing, 32(4), 1615-1635, (2013).
• Dassios, I.K., & Baleanu, D.I. Caputo and related fractional derivatives in singular systems. Applied Mathematics and Computation, 337, 591-606, (2018).
• Dassios, I., Tzounas, G., & Milano, F. Generalized fractional controller for singular systems of differential equations. Journal of Computational and Applied Mathematics, 378, 112919, (2020).
• Dassios, I., & Milano, F. Singular dual systems of fractional-order differential equations. Mathematical Methods in the Applied Sciences, 1-18, (2021).
• Dassios, I., Tzounas, G., & Milano, F. Robust stability criterion for perturbed singular systems of linearized differential equations. Journal of Computational and Applied Mathematics, 381, 113032, (2021).
• Dassios, I.K., Baleanu, D.I., & Kalogeropoulos, G.I. On non-homogeneous singular systems of fractional nabla difference equations. Applied Mathematics and Computation, 227, 112-131, (2014).
• Dassios, I., Kerci, T., Baleanu, D., & Milano, F. Fractional–order dynamical model for electricity markets. Mathematical Methods in the Applied Sciences, 1- 13, (2021).
• Dassios, I., Tzounas, G., Liu, M., & Milano, F. Singular over-determined systems of linear differential equations. Mathematics and Computers in Simulation, 197, 396-412, (2022).
• Gantmacher, R.F. The Theory of Matrices (Vol.1 and Vol.2). Chelsea, New York, (1959).
• Duan, G. R. The Analysis and Design of Descriptor Linear Systems (Vol.23). Springer, (2011).
• Fu, P., Niculescu, S.I., & Chen, J. Stability of linear neutral time-delay systems: Exact conditions via matrix pencil solutions. IEEE Transactions on Automatic Control, 51(6), 1063-1069, (2006).
• Kitano, M., Nakanishi, T., & Sugiyama, K. Negative group delay and superluminal propagation: An electronic circuit approach. IEEE Journal of selected Topics in Quantum electronics, 9(1), 43-51, (2003).
• Lewis, F. L. A survey of linear singular systems. Circuits, Systems and Signal Processing, 5(1), 3-36, (1986).
• Liu, Y., Wang, J., Gao, C., Gao, Z., & Wu, X. On stability for discrete-time non-linear singular systems with switching actuators via average dwell time approach. Transactions of the Institute of Measurement and Control, 39(12), 1771-1776, (2017).
• Liu, M., Dassios, I., & Milano, F. Delay margin comparisons for power systems with constant and time-varying delays. Electric Power Systems Research, 190, 106627, (2021).
• Liu, M., Dassios, I., & Milano, F. On the stability analysis of systems of neutral delay differential equations. Circuits, Systems, and Signal Processing, 38(4), 1639-1653, (2019).
• Liu, M., Dassios, I., Tzounas, G., & Milano, F. Model-independent derivative control delay compensation methods for power systems. Energies, 13(2), 342, (2020).
• Michiels, W., & Niculescu, S.I. Stability and stabilization of time-delay systems: an eigenvalue-based approach. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, (2007).
• Michiels, W., & Niculescu, S.I. Characterization of delay-independent stability and delay interference phenomena. SIAM journal on control and optimization, 45(6), 2138-2155, (2007).
• Milano, F., & Dassios, I. Small-signal stability analysis for non-index 1 Hessenberg form systems of delay differential algebraic equations. IEEE Transactions on Circuits and Systems I: Regular Papers, 63(9), 1521-1530, (2016).
• Milano, F., & Dassios, I. Primal and dual generalized eigenvalue problems for power systems small-signal stability analysis. IEEE Transactions on Power Systems, 32(6), 4626-4635, (2017).
• Naim, M., Sabbar, Y., Zahri, M., Ghanbari, B., Zeb, A., Gul, N., Djilali, S., & Lahmidi, F. The impact of dual time delay and Caputo fractional derivative on the long-run behavior of a viral system with the non-cytolytic immune hypothesis. Physica Scripta, 97(12), 124002, (2022).
• Santra, S.S., Ghosh, A., & Dassios, I. Second-order impulsive differential systems with mixed delays: Oscillation theorems. Mathematical Methods in the Applied Sciences, 45(18), 12184-12195, (2022).
• Tzounas, G., Dassios, I., & Milano, F. Small-signal stability analysis of implicit integration methods for power systems with delays. Electric Power Systems Research, 211, 108266, (2022).
• Tzounas, G., Dassios, I., & Milano, F. Modal participation factors of algebraic variables. IEEE Transactions on Power Systems, 35(1), 742-750, (2019).
• Tzounas, G., Dassios, I., Murad, M.A.A., & Milano, F. Theory and implementation of fractional order controllers for power system applications. IEEE Transactions on Power Systems, 35(6), 4622-4631, (2020).
• Wei, Y., Peter, W.T., Yao, Z., & Wang, Y. The output feedback control synthesis for a class of singular fractional order systems. ISA transactions, 69, 1-9, (2017).
• Yu, X., & Jiang, J. Analysis and compensation of delays in field bus control loop using model predictive control. IEEE Transactions on Instrumentation and Measurement, 63(10), 2432-2446, (2014).